DOUBLE COSET FORMULAS FOR PROFINITE GROUPS

PETER SYMONDS

Abstract. We show that in certain circumstances there is a sort of double coset formula for induction followed by restriction for representations of profinite groups.

1. Introduction

The double coset formula, which expresses the restriction of an induced module as a direct sum of induced modules, is a basic tool in the representation theory of groups, but it is not always valid as it stands for profinite groups. Based on our work on permutation modules in [8], we give some sufficient conditions for a strong form of the formula to hold. These conditions are not necessary, although they do often hold in interesting cases, and at the end we give some simple examples where the formula fails.

We also formulate a weaker version of the formula that does hold in general.

2. Results

We work with profinite groups G and their representations over a complete commutative noetherian local ring R with finite residue class field of characteristic p. There is no real loss of generality in taking R = ℤ̂_{p}. Our representations will be in one of two categories: the discrete p-torsion modules D_{R}(G) or the compact pro-p modules C_{R}(G). These are dual by the Pontryagin duality functor Hom(−, ℚ/ℤ), which we denote by ∗, so we will usually work in D_{R}(G). Our modules are normally left modules, so dual means contragredient. For more details see [7, 8].

For any profinite G-set X and M ∈ D_{R}(G) we let F (X, M) ∈ D_{R}(G) denote the module of continuous functions X → M, with G acting according to (gf)(x) = g(f(g^{−1}x)), g ∈ G, f ∈ F (X, M), x ∈ X. We set F (X) = F (X, T), where T = R^{∗}, the Pontryagin dual of the trivial module R.

When H, G are profinite groups with H ≤ G and M ∈ D_{R}(H), we define the coinduced module Coind_{H}^{G} M ∈ D_{R}(G) by Coind_{H}^{G} M = F (G, M)^{H} ≅ F (G) ˇ⊗^{H}M. Here H acts on the right of G, and G acts on the left, so we have ((g, h)f)(x) = hf(g^{−1}xh) for g, x ∈ G, h ∈ H, f ∈ F (G, M). The operation ˇ⊗ on discrete modules is Pontryagin dual to the more familar completed tensor product ˆ⊗ on compact modules ([8] 2.4); it commutes with direct limits.

The dual concept is induction: for N ∈ C_{R}(H), Ind_{H}^{G} N = R[[G]] ˆ⊗_{H}N, where R[[G]] is the complete group algebra. This is related to coinduction by Pontryagin duality: Ind_{H}^{G} N ≅ (Coind_{H}^{G} N^{∗})^{∗}.

These functors have most of the properties that one would expect by analogy with the discrete case ([8] 2.4).

Partially supported by a grant from the Leverhulme Trust.

Now if K ≤ G then Res_{K}^{G} Coind_{H}^{G} M = Res_{K}^{G}(F (G) ˇ⊗^{H}M) and the action of K is via its action on F (G) only. What we want to do is to decompose F (G), considered as a left K × H-module. The action of K × H on G is given by (k, h)x = kxh^{−1} and the action on F (G) by ((k, h)f)(x) = F (k^{−1}xh), for h ∈ H, k ∈ K, g, x ∈ G.

The reader might prefer to think of the dual problem of decomposing R[[G]], but by working in D_{R}(G) we avoid having to deal with the topology.

What we have is a permutation module in the sense of [8]. Note that Stab_{K×H}(g) = {(k, h) ∈ K × H | kgh^{−1} = g} = {(^{g}h, h) | h ∈ K^{g} ∩ H}.

What we want to do is to decompose this into pieces of the form F ((K × H)/S), where S = S_{g} = Stab_{K×H}(g) for some g ∈ G.

For an H-module M we write gM for the ^{g}H-module which is isomorphic to M as an R-module but where ^{g}h acts as h.

Lemma 2.1. F ((K×H)/S) ˇ⊗^{H}M ≅ Coind_{K∩^{g}H}^{K} Res_{K∩^{g}H}^{H^{g}} gM in D_{R}(K), where S = Stab_{K×H}(g).

Proof. F ((K×H)/S) ˇ⊗^{H}M ≅ F ((K×H)/S, M)^{H} ≅ F (((K×(K^{g}∩H))/S)×_{K^{g}∩H}H, M)^{H} ≅ F ((K × (K^{g} ∩ H))/S, M)^{K^{g}∩H}.

Regard g^{−1}K as a (K × (K^{g} ∩ H))-set by (k_{1}, h_{1})g^{−1}k = h_{1}g^{−1}kk_{1}^{−1}, h_{1} ∈ H, k, k_{1} ∈ K. This action is clearly transitive, and Stab_{K×(K^{g}∩H)}(g) = S, so g^{−1}K ≅ (K × (K^{g} ∩ H)/S). Now the last module in the chain of isomorphisms is isomorphic to F (g^{−1}K, M)^{K^{g}∩H} ≅ F (K, gM)^{K∩^{g}H} ≅ Coind_{K∩^{g}K}^{K} gM. □

This shows that a double coset formula can be obtained as a consequence of a decomposition of a permutation module. For example, we obtain the following easy and well-known result.

Corollary 2.2. If H\G/K is finite then for any M ∈ D_{R}(H) we have the usual double coset formula Res_{K}^{G} Coind_{H}^{G} M ≅ ⨁_{g∈K\G/H} Coind_{K∩^{g}H}^{K} gM and dually for N ∈ C_{R}(H) we have Res_{K}^{G} Ind_{H}^{G} N ≅ ⨁_{g∈K\G/H} Ind_{K∩^{g}H}^{K} gN.

Proof. Since each orbit is closed and there are only finitely many orbits, we see that G ≅ ∐_{g∈K\G/H}(K × H)/S_{g} is a disjoint union of open and closed subspaces, hence F (G) ≅ ⨁_{g∈H\G/K} F ((K × H)/S_{g}). Now use Lemma 2.1 and the discussion preceding it. □

The bewildered reader might wish to consult [1] 3.3.4, where a similar argument is used in the more familiar context of finite groups.

Our more general decomposition of F (G) is based on [8] 3.21, but we will go over the proof again because there the stabilizers were assumed to be finite and we do not want this restriction.

All subgroups and sub-G-sets will be assumed to be closed.

If X is a G-set and Y ≤ X is a sub-G-set then we let F (X, Y) denote the functions on X that are zero on Y. Thus there is a short exact sequence F (X, Y) → F (X) → F (Y). In fact F (X, Y) ≅ F (X/Y, ∗).

For the next four lemmas we will assume the following hypotheses: G is a profinite group, H < G, X is a G-set, Y < X and if x ∈ X − Y then Stab_{G}(x) is conjugate to H in G.

Lemma 2.3. The G-module F (X, Y) is a summand of a direct sum of terms Coind_{H}^{G} T ≅ F (G/H).

Proof. The case of a free action, that is when H = 1, is dealt with in [6] 5.6 when Y = ∅ and in [8] 3.11 otherwise. Basically, the idea is to let N be an open normal subgroup of H and prove the result for the G/N-set X/N; then use the fact that, in this context, “a direct limit of injectives is injective”.

The case of general H is almost the same as in [8] 3.17. If Y ≠ ∅ then F (X, Y) ≅ F (X/Y, ∗) so we can assume that Y = ∗. The case when Y = ∅ is simpler and is left to the reader.

Now apply the case of a free action to the action of N_{G}(H)/H on X^{H} to see that F (X^{H}, ∗) is a summand of a sum of Coind_{1}^{N_{G}(H)/H} T ≅ Coind_{H}^{N_{G}(H)} T ’s. Thus F (G×_{N_{G}(H)}X^{H}, G×_{N_{G}(H)} ∗) = Coind_{N_{G}(H)}^{G} F (X^{H}, ∗) is a summand of a sum of Coind_{H}^{G} T ’s. But the multiplication map (G ×_{N_{G}(H)} X^{H})/(G ×_{N_{G}(H)} ∗) → X is a continuous map of compact Hausdorff sets, hence a homeomorphism, so F (X, ∗) also has this form. □

Lemma 2.4. The short exact sequence F (X^{H}, Y^{H}) → F (X^{H}) → F (Y^{H}) is split over N_{G}(H)/H.

Proof. F (X^{H}, Y^{H}) is an injective module over N_{G}(H)/H, by 2.3. □

Lemma 2.5. The short exact sequence F (X^{H} ∪ Y, Y) → F (X^{H} ∪ Y) → F (Y) is split over N_{G}(H).

Proof. If Y^{H} = ∅ then X^{H} ∪ Y is a disjoint union and the result is clear. Otherwise, the natural map X^{H}/Y^{H} → (X^{H} ∪Y)/Y is a homeomorphism, so F (X^{H} ∪Y, Y) ≅ F (X^{H}, Y^{H}).

Consider the diagram

F (X^{H} ∪ Y, Y) ⟶ F (X^{H} ∪ Y) ⟶ F (Y)
≅↓  ↓  ↓
F (X^{H}, Y^{H}) ⟶ F (X^{H}) ⟶ F (Y^{H}).

The bottom short exact sequence is split, by 2.4, hence the top one is also split. □

Lemma 2.6. The short exact sequence F (X, Y) → F (X) → F (Y) is split over G.

Proof. Let s : F (X^{H} ∪ Y) → F (X^{H} ∪ Y, Y) be the splitting of the previous lemma. Define s′ : F (X) → F (X, Y) by (s′(f))(x) = (s((gf)|_{X^{H}∪Y}))(gx), where f ∈ F (X), x ∈ X and g ∈ G is such that gx ∈ X^{H}.

It is easy to check that s′ is well defined, G-equivariant and a splitting. □

For any two subgroups A, B of G, we write A ≥_{G} B if some G-conjugate of B is a subgroup of A. We write A >_{G} B if B is conjugate to a proper subgroup of A. Since G is compact we can not have A >_{G} A.

Given any G-set X, define:

X^{H} = {x ∈ X| Stab_{G}(x) ≥ H}, X^{>H} = {x ∈ X| Stab_{G}(x) > H},
X^{(H)} = {x ∈ X| Stab_{G}(x) ≥_{G} H}, X^{(>H)} = {x ∈ X| Stab_{G}(x) >_{G} H}.

Clearly X^{H} is closed; so is X^{(H)}, because it is the image of the map G × X^{H} → X, (g, x) ↦ gx. If there are only finitely many conjugacy classes of stabilizers then it is easy to check that X^{>H} and X^{(>H)} are also closed in X.

Theorem 2.7. Let X be a profinite G-set such that there are only finitely many conjugacy classes of stabilizers S_{1}, …, S_{n}. Then F (X) ≅ ⊕_{i=1}^{n}F (X^{(S_{i})}, X^{(>S_{i})}). In particular, F (X) is a summand of a direct sum of the F (X/S_{i})’s.

Proof. We use induction on n; the case n = 0 is trivial. Let S_{1}, say, be minimal among the S_{i} and their conjugates and let Y = ∪_{i=2}^{n}X^{(S_{i})}.

From 2.6 we have F (X) ≅ F (X, Y) ⊕ F (Y). But F (Y) is of the right form by induction and so is F (X, Y) by 2.3. □

Corollary 2.8. Let H, K ≤ G be profinite groups and let K×H act on G by (k, h)g = kgh^{−1}. Suppose that the number of conjugacy classes of stabilizers of the action of K × H on G is finite and let M ∈ D_{R}(H). Then Res_{K}^{G} Coind_{H}^{G} M is a summand of a direct sum of modules of the form Coind_{K∩^{g}H}^{K} gM.

Dually, if N ∈ C_{R}(G) then Res_{K}^{G} Ind_{H}^{G} N is a summand of a direct product of modules of the form Ind_{K∩^{g}H}^{K} gN.

Proof. This follows immediately from 2.7 and 2.1, using K ×H as the group and G as X. □

Remark. If F ⊴ H acts trivially on M then we may replace X = G by X = G/F in the above argument. For example, if M is trivial, so we can take F = H, then we are just considering G/H as a left K-set.

Remark. There are various general conditions that will ensure that the hypotheses of the theorem hold. For example if each K ∩^{g}H is finite then it certainly suffices to know that K × H has only finitely many conjugacy classes of finite subgroups. This is the case if H and K are p-adic analytic by [2]. More generally it holds if H and K are pro-p and virtually of type FP, by [8] 6.15.

For an interesting example of a calculation that uses these ideas see [3, 9].

The statement of the theorem says only a summand of a direct sum of because of the nature of the proof of 2.3. If we know that this module is in fact free then the statement can be made to look more like the usual double coset formula.

Corollary 2.9. In the circumstances of 2.8, let J = K × H and write X for G considered as a J-set as in 2.8. If either both H and K are pro-p groups or if G is countably based then

Res_{K}^{G} Coind_{H}^{G} M ≅ ⨁_{S} F (X^{S}/N_{J}(S), X^{>S}/N_{J}(S)) ˇ⊗ Coind_{K∩^{g}H}^{K} gM,

where S runs through representatives of the conjugacy classes of stabilizers and S is the stabilizer of g ∈ X. The action of K is on the second factor only.

Dually,

Res_{K}^{G} Ind_{H}^{G} N ≅ ∏_{S} R[[X^{S}/N_{J}(S), X^{>S}/N_{J}(S)]] ˆ⊗ Ind_{K∩^{g}H}^{K} gN.

Proof. The proof is the same as that of 2.8 except that we need to identify each term F (X^{(S_{i})}, X^{(>S_{i})}) in the statement of 2.7 as an explicit sum of F (X/S_{i})’s. Let J = K × H and write S for some S_{i}.

Recall that, for a pro-p group P, every projective module is a product of R[[P]]’s by [11] 7.5.4, 7.4.1 or [8] 2.5, so every injective module is a sum of F (P)’s. We can tell how many by considering the fixed point module; if we denote the injective module by I then I ≅ I^{P} ˇ⊗F (P).

But F (X^{(S_{i})}, X^{(>S_{i})})^{J} ≅ (Coind_{N_{J} (S)}^{J} F (X^{S}, X^{>S}))^{J} ≅ F (X^{S}, X^{>S})^{N_{J} (S)} and F (X^{S}, X^{>S}) is an injective module over N_{J}(S)/S by 2.3, so F (X^{S}, X^{>S}) ≅ F (X^{S}, X^{>S})^{N_{J} (S)} ˇ⊗F (N_{J}(S)/S) ∈ D_{R}(N_{J}(S)). Now observe that F (X^{S}, X^{>S})^{N_{J} (S)} ≅ F (X^{S}/N_{J}(S), X^{>S}/N_{J}(S)).

If G is countably based then so is X^{S}, so the quotient map X^{S} → X^{S}/N_{J}(S) has a continuous section s by [6] 5.6.7. But then the map Im(s) × N_{J}(S)/S → X^{S} induces an isomorphism F (X^{S}, X^{>S}) ≅ F ((X^{S}/N_{J}(S)) × N_{J}(S)/S, (X^{>S}/N_{J}(S)) × N_{J}(S)/S) ≅ F (X^{S}/N_{J}(S), X^{>S}/N_{J}(S)) ˇ⊗F (N_{J}(S)/S).

In either case F (X) ≅ ⊕_{S}F (X^{S}/N_{J}(S), X^{>S}/N_{J}(S)) ˇ⊗F (J/S), and hence the claim. □

3. The General Case

A completely different approach to these formulas is given by Mel’nikov [4, 5]. He succeeds in expressing R[[X]] ≅ ⨁_{x∈G\X} R[[G/ Stab_{G}(x)]], where the ⊕ is not normally the usual direct sum, but some sort of completion of it. For example, if G is the trivial group then we have R[[X]] ≅ ⨁_{x∈X} R.

This ⊕ does, however, have the good property of commuting with Tor.

By the discussion in section 2, we can now write Res_{K}^{G} Ind_{H}^{G} M ≅ ⨁_{g∈H\G/K} Ind_{H^{g}∩K}^{K} g^{−1}M, suitably interpreted, for any H, K ≤ G.

We now present a result that is essentially dual to this.

Recall that for any N ∈ C_{R}(G), M ∈ D_{R}(G) there are well-behaved Galois Ext-groups Ext_{G}^{n}(N, M) (see [6] 6.1, [8] 2.6).

For any left G-set X let O_{x}(G, X) = Gx denote the orbit of x ∈ X. By abuse of notation we will often allow x ∈ X/G and write just O_{x}. The inclusion of O_{x} in X induces a restriction map r_{x} : Ext_{G}^{n}(N, F (X)) → Ext_{G}^{n}(N, F (O_{x})) for any N ∈ C_{R}(G).

We give all the groups Ext_{G}^{n}(N, F (O_{x})) the discrete topology and form their direct product, giving it the product topology. The r_{x} combine to induce a map θ : Ext_{G}^{n}(N, F (X)) → ∏_{x∈X/G} Ext_{G}^{n}(N, F (O_{x})).

When we take n = 0 and N = R[[G]] this reduces to the more basic homomorphism θ : F (X) → ∏_{x∈X/G} F (O_{x}).

Theorem 3.1. The map θ is injective with dense image.

Proof. The map θ is certainly an isomorphism when X is finite.

In order to simplify the notation we write E(−) for Ext_{G}^{n}(N, F (−)). Write X = \underleftarrow{lim} X_{i}, an inverse limit of finite G-sets indexed by some set I (this is always possible by [6] 5.6.4). Thus there are G-equivariant morphisms p_{i} : X → X_{i} and we say that X_{j} covers X_{i} if there is a morphism X_{j} → X_{i} in the system. We know that E(X) ≅ \underrightarrow{lim} E(X_{i}).

For any a ∈ E(X) there is an s ∈ I and an a_{s} ∈ E(X_{s}) such that a = p_{s}^{∗}a_{s}. Suppose that θ(a) = 0 and consider θ(a_{s}).

Given x ∈ X, the restriction of p_{s} to O_{x}(G, X) yields a map E(O_{p_{s}(x)}(G, X_{s})) → E(O_{x}(G, X)), under which the image of r_{p_{s}(x)}a_{s} is zero. Thus for each x ∈ X there is some X_{i}, say X_{i(x)}, that covers X_{s} and such that the image of r_{p_{s}(x)}a_{s} under the natural map E(O_{p_{s}(x)}(G, X_{s})) → E(O_{p_{i(x)}(x)}(G, X_{i(x)})) is zero.

Let Y_{x} = p_{i(x)}^{−1}(O_{p_{i(x)}(x)}(G, X_{i(x)})). The Y_{x}, x ∈ X provide an open cover of X so, by compactness, there is an open subcover Y_{i_{1}}, …, Y_{i_{u}}. Let X_{t} be a set that covers all of X_{i_{1}}, … X_{i_{u}}.

By construction, the image of θ(a_{s}) in ∏_{x∈X_{t}/G} E(O_{x}(G, X_{t})) is zero. It follows that the image of a_{s} in E(X_{t}) must be zero, so a = 0, proving injectivity.

For the density statement, notice that it just means that, given a finite set of disjoint orbits O_{x_{1}}, …, O_{x_{v}} and elements a_{t} ∈ E(O_{x_{t}}), we can find an a ∈ E(X) such that r_{x_{t}}(a) = a_{t} for t = 1, …, v.

There must be some X_{i}, say X_{k}, such that the orbits O_{x_{t}}(G, X_{k}) for t = 1, …, v are distinct. Also, for each x_{t} there is an X_{i}, say X_{i(t)}, such that a_{t} is the image of some ā_{t} ∈ E(O_{p_{i(t)}(x_{t})}(G, X_{i(t)})). Let X_{w}, say, cover X_{k} and all the X_{i(t)} and let ã_{t} be the image of ā_{t} in E(O_{p_{w}(x_{t})}(G, X_{w})).

Certainly there is an element ã ∈ E(X_{w}) such that r_{p_{w}(x_{t})}(ã) = ã_{t} for each t, so if a denotes the image of ã in E(X) then r_{x_{t}}(a) = a_{t} for t = 1, …, v, as required. □

Corollary 3.2. Whenever we have subgroups H, K ⊆ G, the K-equivariant map

F (G/H) → ∏_{g∈K\G/H} F (O_{g}(K, G/H)) ≅ ∏_{x∈K\G/H} F (K/(K ∩^{g}H))

and, for N ∈ C_{R}(K), M ∈ D_{R}(H), the map

Ext_{K}^{n}(N, Coind_{H}^{G} M) → ∏_{g∈K\G/H} Ext_{K}^{n}(N, F (O_{g}(K, G/H)) ˇ⊗^{H}M) ≅ ∏_{g∈K\G/H} Ext_{K∩^{g}H}^{n}(N, gM).

are injective with dense image.

Proof. The first part follows immediately from 3.1 and 2.1 in the same way as 2.8.

For the second part we also consider G as a K × H-set, but first let X be an arbitrary K × H-set on which H acts freely and consider Ext_{K×H}^{n}(N ˆ⊗M^{∗}, F (X)).

Since H acts freely, F (X) is injective over H. So a standard spectral sequence argument shows that Ext_{K×H}^{n}(N ˆ⊗M^{∗}, F (X)) ≅ H^{n}(K, Hom_{H}(N ˆ⊗M^{∗}, F (X))) and in turn this is isomorphic to H^{n}(K, Hom(N, Hom_{H}(M^{∗}, F (X)))) ≅ Ext_{K}^{n}(N, Hom_{H}(M^{∗}, F (X))) ≅ Ext_{K}^{n}(N, F (X) ˇ⊗^{H}M).

If we do take X = G then this becomes Ext_{K}^{n}(N, Coind_{H}^{G} M). But we can also decompose F (X) into orbits first and we then obtain the terms in the middle direct product. These are isomorphic to those in the right hand product by 2.1 and Shapiro’s Lemma. Now apply 3.1. □

Remark. A theorem for which this result plays a key role in the proof can be found in [10].

4. Example

Let G be the group of matrices of the form [\begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}] with a ∈ 𝔽_{p} and b, c ∈ 𝔽_{p}[[T]]. This

acts on V = 𝔽_{p}[[T]]^{2}, considered as column vectors [\begin{matrix}x \\ y \\ 1 \end{matrix}]. This action is transitive and the

stabilizer of [\begin{matrix}x \\ y \\ 1 \end{matrix}] consists of those matrices with c = 0 and b = −ay, so is cyclic of order p.

Let H consist of those matrices with b = c = 0: then V ≅ G/H. Let K consist of those matrices with c = 0 (an infinite elementary abelian p-group) and consider V as a K-set.

Let F denote the continuous functions with values in 𝔽_{p}. Suppose that some F (K/S) is a summand of F (V) as a K-module, where S is a stabilizer of some point of V. Then because of the action of G, which conjugates the S’s, every F (K/S) is a summand. These are pairwise non-isomorphic and their endomorphism rings are local, by [9]. Thus, by the proof of the Krull-Schmidt property, any finite direct sum of distinct F (K/S)’s is a summand of F (V), hence the full direct sum ⊕_{S}F (K/S) is a submodule of F (V). But F (V) is countable and the indexing set of the S’s is not: a contradiction.

Thus there can be no formula of the type discussed above for Res_{K}^{G} Coind_{H}^{G} 𝔽_{p}. There is a similar example with a, b, c ∈ ℤ̂_{p}. In this case K ≅ ℤ̂_{p}^{2} and each S ≅ ℤ̂_{p}.

References

[1] Benson, D. J., Representations and Cohomology I, Cambridge Univ. Press. 1992.

[2] J.D. Dixon, M.P.F. du Sautoy, A. Mann and D. Segal, Analytic pro-p groups (2nd edition), London Math. Soc. Lecture Notes 157 (1999), Cambridge Univ. Press, Cambridge.

[3] P. Goerss, H.-W. Henn, M. Mahowald and C. Rezk, A resolution of the K(2)-local sphere at the prime 3, Annals Math. 162 (2005) 777-822.

[4] O.V. Mel’nikov, Subgroups and homology of free products of profinite groups, Math. USSR Izvestiya 34 (1990) 97-119.

[5] O.V. Mel’nikov, Aspherical pro-p-groups, Mat. Sbornik 193 (2002) 71-104.

[6] L. Ribes and P. Zalesskii, Profinite Groups, Springer-Verlag, Berlin 2000.

[7] P. Symonds and T. Weigel, Cohomology of p-adic analytic groups, in New Horizons in pro-p groups, du Sautoy, Segal, Shalev, eds., Progress in Math. 184 (2000), Birkhäuser, Boston.

[8] P. Symonds, Permutation complexes for profinite groups, Comment. Mathematici Helvetici 82 (2007) 1-37.

[9] P. Symonds, On the construction of permutation complexes for profinite groups, Geometry and Topology Monographs 11 (2007) 369-378.

[10] P. Symonds, On cohomology isomorphisms of groups, Jour. Algebra 313 (2007) 802-810.

[11] J.S. Wilson, Profinite Groups, Oxford Univ. Press, 1998.

School of Mathematics, University of Manchester, P.O. Box 88, Manchester M60 1QD, England

 E-mail address: Peter.Symonds@manchester.ac.uk